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A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called "effective equation" including the reduced integral kernel (called "effective kernel") in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations. (C) 2020 The Authors. Published by Elsevier Ltd.

In this paper, we study the existence of traveling wave solutions connecting two constant states to a nonlocal scalar equation with sign-changing kernel. A typical example of such kernel in the neural fields is the Mexican hat type function. We first introduce a new notion of upper-lower-solution for the equation of wave profile for a given wave speed. Then, with the help of Schauder's fixed point theorem, we construct two different pairs of upper-lower-solutions to obtain traveling waves for a continuum of wave speeds under two different assumptions. Due to the sign-changing nature of the kernel, the wave profiles may take both positive and negative values. Finally, we analyze the limit of the right-hand tail of wave profiles. Under some further condition on the wave speeds, we prove that the right-hand tail limit of the wave profile does exist. In particular, we obtain the existence of nonnegative traveling waves connecting the unstable state 0 and the stable state 1 for wave speeds large enough.

We explored the use of ecological niche-modeling from data compiled in a participatory program of thousands of volunteer observers to quantitatively evaluate the spatial risk of invasion by the alien species Bombus terrestris. B. terrestris has been commercially introduced to Japan for the pollination of greenhouse tomato crops since 1991. Recently, naturalized populations of this species have spread rapidly, particularly in Hokkaido, northern Japan. Two niche-based models were constructed from independent data sets (presence–absence and presence-only data sets) and were used to validate one another. Both models performed well and indicated that the data compiled by participatory programs were useful for generating predictive models of the potential distribution of this invasive species. The models also revealed that the potential distribution of B. terrestris is negatively related to the proportion of woodland area, which corresponds to the known biology of this species. In contrast, the effect of the number of colonies introduced for tomato pollination (represented by tomato production) did not significantly affect the distribution of B. terrestris, indicating that its spread is now entering a phase of dispersal from established feral populations and that primary dispersal from commercial colonies in greenhouses is no longer a limiting factor for the distribution of this invasive species.

Studies of the interspecific relationships between alien and native pollinator species can help forecast the success of alien species as well as assess the extent of disturbance to native plant–pollinator interactions. We examined the habitat and flower resource occupancy by the invasive introduced bumble bee Bombus terrestris and three dominant native bumble bees in central Hokkaido, northern Japan, in relation to a landscape factor (forest cover ratio) and flower morphology with respect to the proboscis length of bees. Three years of monitoring the invasive boundary of B. terrestris indicated that this species, which dominates open agricultural areas, probably will not invade the forests in which the native species dominate. This habitat partitioning likely followed the displacement of the natives by the invader in open agricultural lands. In forested areas, the native species partitioned flower resources on the basis of the relationship between proboscis length and the lengths of the corolla tubes of flowers. However, in open agricultural areas, both the long- and short-tubed flowers were primarily visited by the exotic short-tongued B. terrestris, which foraged illegitimately (by nectar robbing) on long-tubed flowers (Trifolium pratense L.) and legitimately on short-tubed flowers (T. repens L. and Lavandula angustifolia Mill.). The invasion of B. terrestris into open natural vegetation, in addition to open agricultural areas, has recently been reported in Hokkaido. Even though the exotic and dominant native bumble bees partition their habitat according to landscape factors, the invasive bee has the potential to alter the overall interactions within plant–pollinator systems in the regions of open vegetation on Hokkaido Island.